Crack CAT Quantitative Aptitude by learning the concept of Variation with Examples

DIRECT VARIATION

• Quantity A is said to vary directly as another Quantity B when the two quantities A and B depend on each other in such a manner that if Quantity B is changed, then Quantity A changes in the same ratio.

• The symbol used for variation is ‘α ’

• Also, if A α B(spoken as: A directly varies as B), then there exists a positive constant m, called the constant of proportionality, such that A = mB

Comments:

A varies directly as B means that the ratio of A to B is constant. That is, if
a₁, a₂, a₃ … and
b₁, b₂, b₃ …
are corresponding values of A and B, respectively, then Adirectly varies as B means:

i.e. this constant ratio is m.
Thus ai/bi = m or ai = mbi for each i. We write this as A = mB. Note that m is a positive real number.

INDIRECT VARIATION

• When A varies inversely as B, we write

, then there exists a positive constant k such that AB = k.
That is, if A varies inversely as B means that .

Thus, there exists a constant k such that

This statement is the converse of the statement made under Direct Variation.

Comments:

A varies indirectly as B means that the product AB is equal to a constant. That is, if

a1, a2, a3 … and

b1, b2, b3 …

are corresponding values of A and B, respectively, then Aindirectlyvaries as B means:

aibi = k for each i. We write this as AB = k. where k is a positive real number.

OTHER PROPERTIES OF VARIATION

• If A varies directly as B, then B varies directly as A.

Comments: If A = mB and m is positive, then . Thus B α A.

So we can say “A and B very directly” with each other.

• If A varies inversely as B, then B also varies inversely as A.  

Comments:

If AB = k, then BA = k. So 

So, we can say “A and B vary inversely” with respect to each other.

• If A varies jointly with B, C, …, Z means that A BC …Z

       A varies jointly with B, C, …, D and inversely with F, G, …, H means

This is also known as Combined Variation.

Illustration and Application of Combined Variation

Which of the following equations in the given options correctly represents the phenomenon of gravitational force F, if m1 and m2 are the masses of the bodies between which the force of gravity exists and d is the distance between them? It is known that F varies directly as the masses and inversely as the square of the distance between them.

It is known that F varies directly as the masses and inversely as the square of the distance between them.

In the problem above,

  we have F varying directly with masses m1 and m2 i.e. F α m1 m2

  and also varying indirectly with

therefore we can write

or